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Moments of Force D. Gordon E. Robertson, PhD, FCSB Biomechanics Laboratory, School of Human Kinetics, University of Ottawa, Ottawa, Canada D. Gordon E. Robertson, PhD, FCSB Biomechanics Laboratory, School of Human Kinetics, University of Ottawa, Ottawa, Canada

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Moment of a Force (when F is at 90º to d ) turning effect of a force, also called torque product of force (F) and moment arm (d) of the force from the axis ( A ) of rotation moment arm is the perpendicular distance from the axis of rotation to the line of the force M = F d force ( F ) axis (A) line of action of force moment arm (d)

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Moment of a Force (when F is at 90º to d ) M = F d direction (+ / − sign) of the moment of force depends on the right- hand rule i.e., counter-clockwise is positive units are newton metres or N.m force ( F ) axis (A) line of action of force moment arm (d)

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Moment of a Force ( r, F, ) if moment arm length is difficult to compute use: M = r F sin r is length of line from axis to line of force theta is angle between line of force and line of r force ( F ) axis (A) line of action of force line from A to force (r) angle between r and F ( )

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Moment of a Force ( r, F, ) M = r F sin direction (sign) of moment follows right- hand rule i.e., if force “turns” line r counter- clockwise about axis at A then moment is positive force ( F ) axis (A) line of action of force line from A to force (r) angle between r and F ( )

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Moment of a Force ( r, F, ) to simplify and clarify positive directions of moments and forces, add reference axes to each figure force ( F ) axis (A) line of action of force line from A to force (r) angle between r and F ( ) + positive directions are defined by the arrows add axes labels (X, Y) for additional clarity Y X

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Moment of a Force ( r, F, ) Factors that increase moment of force increase force (F) increase lever arm length (r) increase angle ( ) between lever and line of force to perpendicular M = r F sin force ( F ) axis (A) line of action of force line from A to force (r) angle between r and F ( )

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Moment of a Force (cross-product) if components of force and line connecting axis to line of force are known, use vector cross-product: M = r x F force ( F ) axis (A) r

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Moment of a Force (cross-product) first resolve vectors r and F into their rectangular coordinates then apply: M = r x F = ( r x F y − r y F x ) k k is the unit vector about the Z-axis axis (A) force ( F ) axis (A) r

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Moment of a Force (cross-product) first resolve vectors r and F into their rectangular coordinates then apply: M = r x F = ( r x F y − r y F x ) k k is the unit vector about the Z-axis force ( F ) axis (A) r FyFy FxFx ryry rxrx

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Moment of a Force (cross-product) put components at their original points of application force ( F ) axis (A) FyFy FxFx ryry rxrx

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Moment of a Force (cross-product) put components at their original points of application next slide force vectors along their lines of action and multiply force ( F ) axis (A) r FyFy FxFx ryry rxrx

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Moment of a Force (cross-product) if only scalar part of the moment is wanted, use this notation: M = [ r x F ] z M = ( r x F y − r y F x ) force ( F ) axis (A) r FyFy FxFx ryry rxrx

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Example Given: r = (20.0, −65.0) cm F = (220, 150.0) N M = [ r x F y − r y F x ] z =[ (20.0 150.0) − (-65.0 220) ] = 3000 + 14 300 = 17 300 N.cm = 173.0 N.m force ( F ) axis (A) r FyFy FxFx ryry rxrx

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